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Progress Toward a Robotic Bat:
Modeling, Design, Dynamics, and Control
Alireza Ramezani,
Xichen Shi,
Jon Hoff,
Soon-Jo Chu...
Swartz & Breuer,
Co-PI’s at Brown University
Bio-inspired
flapping flight
Safe
Quiet
Energy efficient
Overview
 Objectives
 Robot Morphology
 Reduced-order flight apparatus
 Modeling
 Trajectory planning
 Nonlinear and...
B2: Robot Morphology
Five Degrees of Actuation:
 Wings flapping motion
 Wing folding and unfolding
 Legs dorsoventral m...
B2: Robot Morphology
Articulated Flight Mechanism
Reduced-order flight apparatus
 Biologically meaningful DoFs
 Retraction-protraction
 Flexion-extension
 Abduction-add...
B2: Robot Morphology
Autonomy
Computer
Encoders
Power
Electronics
IMU
Data Storage
Bat Configuration
Pose, Actuated and Unactuated Joints
𝑞 = (𝑥, 𝑦, 𝑧, 𝜙, 𝜃, 𝜓, 𝑞 𝑝, 𝑞 𝑎)
Center of Mass
Position
Body Frame...
Nonlinear Model
Aerodynamic and constraint forces
ℳ 𝑞 𝑞 + 𝒞 𝑞, 𝑞 𝑞 + 𝒢 𝑞 = 𝒬 𝜆 + 𝒬ℱ
𝒬ℱ = Generalized Aerodynamic Forces
𝒬 ...
Nonlinear Model
In unactuated and actuated coordinates
We can now partition the system dynamics into unactuated
coordinate...
Trajectory planning
Parametrized manifolds
• Define 𝑟𝑑 𝑡, 𝛽 as the desired trajectory of the actuated joints.
• The desire...
Trajectory planning
Constrained dynamics
Parametric Manifold:
𝒩 = 0
with
𝒩 = 𝑞 𝑎 − 𝑟𝑑(𝑡, 𝛽)
We now have two equations gove...
Trajectory planning
Constraint manifold revisited
• To ameliorate numerical issues, we consider position, velocity and
acc...
Trajectory planning
Constrained dynamics revisited
Parametric Manifold:
𝑞 𝑎 =
𝜕𝒩
𝜕𝑞 𝑎
−1
−𝜅1 𝒩 − 𝜅2 𝒩 −
𝜕2 𝒩
𝜕𝑡2
−
𝜕
𝜕𝑞 𝑎
...
Trajectory planning
Gradient-based optimization
Given the constrained system dynamics, we use a shooting-based
approach to...
can be written as
ℳ1 𝜒 + 𝒞1 𝜒 = −ℳ2 𝑞 𝑎 − 𝒞2 𝑞 𝑎 − 𝒢1 + 𝒬ℱ,1
since the Jacobian is zero for the unactuated degrees of free...
Nonlinear Control
Non-classic tracking problem
When the constraints 𝒩 are satisfied, we have
𝑞 𝑎 =
𝜕𝑟𝑑
𝜕𝑡
+
𝜕𝑟𝑑
𝜕𝛽
𝛽
𝑞 𝑎 =...
Nonlinear Control
Contraction theory
These equivalent dynamics converge exponentially to a predefined
desired trajectory 𝜒...
7-DoF Flapping Model
Tailless flapping robot
Degrees of Freedom
 Flapping
 Feathering
 Parasagital pitch
 Vertical Ele...
Right flapping angle
7-DoF Flapping Model
Parametric trajectory
Doman, Oppenheimer, Bolender, Sightorsson, “Attitude
contr...
7-DoF Flapping Model
Simulation
Pitchangletracking
thedesiredangle
Featheringangle
Aerodynamic forces Center of pressure v...
7-DoF Flapping Model
Simulation
This approach relies on a parameterized desired trajectory, and fairly
simple wing kinemat...
Bio-Inspired Design
Simultaneous optimization of kinematic design and wing trajectory
The Challenge:
Design a simplified k...
𝑞RP
𝑞FE
𝑞AA
𝑞DV
𝑥s
𝑞FL
𝑙
Optimize link lengths
and angle offsets
Robot Bat
Hind
Limb
Wing
Optimize joint
trajectories
Kinematic Parameters
𝑞RP
𝑞FE
𝑞AA
𝑞DV
𝑥s
𝑞FL
𝛾1
𝛾2
𝛾3
𝛾4
ℎ1
ℎ2
ℎ3
𝑟2
𝑑1
𝑑2
𝑑3
𝑟
𝑙
𝑏
𝑦
𝑦
Actuated DOF in RED
Black are the r...
Optimization
Iterative optimization of flapping parameters and kinematic parameters
Iteratively optimize
 flapping angle ...
Preliminary Results
• Solid diamonds are biological bat
data.
• Open circles are for the optimized
robot bat design.
Futur...
Concluding Remarks
 Kinematic design – passive and active joints
 Nonlinear dynamics – many unactuated degrees of freedo...
Undergraduate Researchers
Graduate Researchers
Principal InvestigatorsPostdoc
Upcoming SlideShare
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Seth Hutchinson - Progress Toward a Robotic Bat

2015 NSF NRI PI Meeting

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Seth Hutchinson - Progress Toward a Robotic Bat

  1. 1. Progress Toward a Robotic Bat: Modeling, Design, Dynamics, and Control Alireza Ramezani, Xichen Shi, Jon Hoff, Soon-Jo Chung, Seth Hutchinson Jon Hoff (Ph.D student)
  2. 2. Swartz & Breuer, Co-PI’s at Brown University Bio-inspired flapping flight Safe Quiet Energy efficient
  3. 3. Overview  Objectives  Robot Morphology  Reduced-order flight apparatus  Modeling  Trajectory planning  Nonlinear and non-affine-in-control tracking problem  Results  Simultaneous kinematics/trajectory design  Future works & conclusion Maybe a bit overly optimistic...
  4. 4. B2: Robot Morphology Five Degrees of Actuation:  Wings flapping motion  Wing folding and unfolding  Legs dorsoventral movement
  5. 5. B2: Robot Morphology Articulated Flight Mechanism
  6. 6. Reduced-order flight apparatus  Biologically meaningful DoFs  Retraction-protraction  Flexion-extension  Abduction-adduction Leg Mechanism:  Elevation-depression
  7. 7. B2: Robot Morphology Autonomy Computer Encoders Power Electronics IMU Data Storage
  8. 8. Bat Configuration Pose, Actuated and Unactuated Joints 𝑞 = (𝑥, 𝑦, 𝑧, 𝜙, 𝜃, 𝜓, 𝑞 𝑝, 𝑞 𝑎) Center of Mass Position Body Frame Orientation (Euler Angles) Passive Joints Active Joints 𝜒 = 𝑥, 𝑦, 𝑧, 𝜙, 𝜃, 𝜓, 𝑞 𝑝 𝑇 It will be convenient to bundle the unactuated degrees of freedom: 𝑞 = 𝜒 𝑞 𝑎
  9. 9. Nonlinear Model Aerodynamic and constraint forces ℳ 𝑞 𝑞 + 𝒞 𝑞, 𝑞 𝑞 + 𝒢 𝑞 = 𝒬 𝜆 + 𝒬ℱ 𝒬ℱ = Generalized Aerodynamic Forces 𝒬 𝜆 = 𝜕𝑞 𝑎 𝜕𝑞 𝑇 𝜆 = Constraint Forces (kinematic) 𝜕𝑞 𝑎 𝜕𝑞 = Jacobian for actuated joints = [0 | 𝐼 𝑁 𝑎×𝑁 𝑎 ]
  10. 10. Nonlinear Model In unactuated and actuated coordinates We can now partition the system dynamics into unactuated coordinates 𝜒 and actuated coordinates 𝑞 𝑎: ℳ1 ℳ2 ℳ3 ℳ4 𝜒 𝑞 𝑎 + 𝒞1 𝒞2 𝒞3 𝒞4 𝜒 𝑞 𝑎 + 𝒢1 𝒢2 = 𝜕𝑞 𝑎 𝜕𝑞 𝑇 𝜆 + 𝒬ℱ which can be written as: 𝜒 𝑞 𝑎 = ℳ1 ℳ2 ℳ3 ℳ4 −1 − 𝒞1 𝒞2 𝒞3 𝒞4 𝜒 𝑞 𝑎 − 𝒢1 𝒢2 + 𝜕𝑞 𝑎 𝜕𝑞 𝑇 𝜆 + 𝒬ℱ
  11. 11. Trajectory planning Parametrized manifolds • Define 𝑟𝑑 𝑡, 𝛽 as the desired trajectory of the actuated joints. • The desired trajectory is parameterized by 𝛽, which includes, e.g., flapping frequency, flapping amplitude, etc. • If the actuated joints follow the desired trajectory, the system state will evolve on the manifold: 𝑆 = 𝑞, 𝑞 ∈ 𝒯𝒬 | 𝒩 = 0 with 𝒩 = 𝑞 𝑎 − 𝑟𝑑(𝑡, 𝛽) Our design goal is to define the desired trajectory 𝑟𝑑(𝑡, 𝛽) so that trajectories in 𝑆 satisfy appropriate performance criteria.
  12. 12. Trajectory planning Constrained dynamics Parametric Manifold: 𝒩 = 0 with 𝒩 = 𝑞 𝑎 − 𝑟𝑑(𝑡, 𝛽) We now have two equations governing the system dynamics. Lagrangian Dynamics: 𝜒 𝑞 𝑎 = ℳ1 ℳ2 ℳ3 ℳ4 −1 − 𝒞1 𝒞2 𝒞3 𝒞4 𝜒 𝑞 𝑎 − 𝒢1 𝒢2 + 𝜕𝑞 𝑎 𝜕𝑞 𝑇 𝜆 + 𝒬 ℱ This formulation is prone to numerical difficulties due to computations of derivatives...
  13. 13. Trajectory planning Constraint manifold revisited • To ameliorate numerical issues, we consider position, velocity and acceleration constraints: 𝒩 + 𝜅1 𝒩 + 𝜅2 𝒩 = 0 • Computing derivatives (applying the chain rule w.r.t. 𝑞 𝑎), we obtain: 𝑞 𝑎 = 𝜕𝒩 𝜕𝑞 𝑎 −1 −𝜅1 𝒩 − 𝜅2 𝒩 − 𝜕2 𝒩 𝜕𝑡2 − 𝜕 𝜕𝑞 𝑎 𝜕𝒩 𝜕𝑞 𝑎 𝑞 𝑎 𝑞 𝑎 This gives an expression for the dynamics of 𝑞 𝑎 in terms of the desired trajectory 𝑟𝑑 𝑡, 𝛽 .
  14. 14. Trajectory planning Constrained dynamics revisited Parametric Manifold: 𝑞 𝑎 = 𝜕𝒩 𝜕𝑞 𝑎 −1 −𝜅1 𝒩 − 𝜅2 𝒩 − 𝜕2 𝒩 𝜕𝑡2 − 𝜕 𝜕𝑞 𝑎 𝜕𝒩 𝜕𝑞 𝑎 𝑞 𝑎 𝑞 𝑎 with 𝒩 = 𝑞 𝑎 − 𝑟𝑑(𝑡, 𝛽) We now have two equations governing the system dynamics. Lagrangian Dynamics: 𝜒 𝑞 𝑎 = ℳ1 ℳ2 ℳ3 ℳ4 −1 − 𝒞1 𝒞2 𝒞3 𝒞4 𝜒 𝑞 𝑎 − 𝒢1 𝒢2 + 𝜕𝑞 𝑎 𝜕𝑞 𝑇 𝜆 + 𝒬 ℱ Given a parameterized trajectory 𝑟𝑑 𝑡, 𝛽 , 𝛽, and initial conditions 𝑞 𝑡0 , 𝑞(𝑡0), we can compute the evolution of the unactuated states 𝜒.
  15. 15. Trajectory planning Gradient-based optimization Given the constrained system dynamics, we use a shooting-based approach to optimize the choice of desired trajectory: • Enforce periodicity in appropriate coordinates of 𝑞. • Ensure stability. • Ensure that the unactuated coordinates 𝜒 satisfy constraints on flight performance. • Minimize expended energy. This is achieved by • numerically integrating the system dynamics • numerically calculating the gradient of the objective function to find periodic and feasible solutions → Matlab’s fmincon
  16. 16. can be written as ℳ1 𝜒 + 𝒞1 𝜒 = −ℳ2 𝑞 𝑎 − 𝒞2 𝑞 𝑎 − 𝒢1 + 𝒬ℱ,1 since the Jacobian is zero for the unactuated degrees of freedom. The unactuated system dynamics given by the top row of ℳ1 ℳ2 ℳ3 ℳ4 𝜒 𝑞 𝑎 + 𝒞1 𝒞2 𝒞3 𝒞4 𝜒 𝑞 𝑎 + 𝒢1 𝒢2 = 0 𝐼 𝜆 + 𝒬ℱ,1 𝒬ℱ,2 Nonlinear Control The unactuated system dynamics The trajectory design gives a desired trajectory for both 𝜒 and 𝑞 𝑎. Since 𝑞 𝑎 concerns the actuated coordinates, we formulate our tracking problem in terms of the unactuated coordinates 𝜒.
  17. 17. Nonlinear Control Non-classic tracking problem When the constraints 𝒩 are satisfied, we have 𝑞 𝑎 = 𝜕𝑟𝑑 𝜕𝑡 + 𝜕𝑟𝑑 𝜕𝛽 𝛽 𝑞 𝑎 = 𝜕2 𝑟𝑑 𝜕𝑡2 + 𝜕 𝜕𝛽 𝜕𝑟𝑑 𝜕𝛽 𝛽 𝛽 + 𝜕𝑟𝑑 𝜕𝛽 𝛽 𝑞 𝑎 = 𝑟𝑑 We use a calculus of variations approach to update 𝛽, to give the resulting equivalent dynamics: ℳ1 𝜒 + 𝒞1 𝜒 = −ℳ2 𝜒 𝑟 − 𝒞2 𝜒 𝑟 − 𝜅 𝜒 − 𝜒 𝑟 with 𝜅 a positive definite gain matrix, and 𝜒 𝑟 a defined reference trajectory. The actuated coordinates in terms of control input 𝛽
  18. 18. Nonlinear Control Contraction theory These equivalent dynamics converge exponentially to a predefined desired trajectory 𝜒 𝑑 when 𝜒 𝑟 = 𝜒 𝑑 − Λ 𝜒 − 𝜒 𝑑 since the equivalent dynamics can be written as ℳ1 𝑠 + 𝒞1 𝑠 + 𝜅𝑠 = 0 for 𝑠 = 𝜒 − 𝜒 𝑟 From trajectory planning optimization
  19. 19. 7-DoF Flapping Model Tailless flapping robot Degrees of Freedom  Flapping  Feathering  Parasagital pitch  Vertical Elevation (z)  Horizontal Position (x) Control Inputs
  20. 20. Right flapping angle 7-DoF Flapping Model Parametric trajectory Doman, Oppenheimer, Bolender, Sightorsson, “Attitude control of a single degree of freedom flapping wing micro air vehicle,” AIAA GNC Conf. 2009. 𝛽 − Flapping frequency and amplitude, feathering angle, etc. Right feathering angle
  21. 21. 7-DoF Flapping Model Simulation Pitchangletracking thedesiredangle Featheringangle Aerodynamic forces Center of pressure velocities
  22. 22. 7-DoF Flapping Model Simulation This approach relies on a parameterized desired trajectory, and fairly simple wing kinematics... Perhaps we could do better
  23. 23. Bio-Inspired Design Simultaneous optimization of kinematic design and wing trajectory The Challenge: Design a simplified kinematic model that is capable of achieving wing shapes similar to biological bat throughout wingbeat cycle. The biological bat has many degrees of freedom, but it is not yet completely understood how these degrees of freedom are coupled to effect wing motion. Motion capture data can be analyzed to obtain a principal component analysis of the wing shape. This facilitates comparison between robot and biological bat.P. Chen, S. Joshi, S. Swartz, K. Breuer, J. Bahlman, G. Reich, “Bat Inspired Flapping Flight,” AIAA SciTech, 2014.
  24. 24. 𝑞RP 𝑞FE 𝑞AA 𝑞DV 𝑥s 𝑞FL 𝑙 Optimize link lengths and angle offsets Robot Bat Hind Limb Wing Optimize joint trajectories
  25. 25. Kinematic Parameters 𝑞RP 𝑞FE 𝑞AA 𝑞DV 𝑥s 𝑞FL 𝛾1 𝛾2 𝛾3 𝛾4 ℎ1 ℎ2 ℎ3 𝑟2 𝑑1 𝑑2 𝑑3 𝑟 𝑙 𝑏 𝑦 𝑦 Actuated DOF in RED Black are the results of movement Grey are constant angles Joint Variables: • 𝑞RP: Retraction-protraction (shoulder angle) • 𝑞FE: flexion-extension (elbow angle) • 𝑞AA: abduction-adduction (wrist angle) • 𝑞FL: flapping angle wrt. xy plane • 𝑞DV: dorsoventral movement of hind limb • 𝑥: spindle position Ten Link Parameters: • ℎ1: humerus • ℎ2: humerus • ℎ3: humerus • 𝑟2: radius • 𝑟: wrist • 𝑑1: finger 1 length • 𝑑2: finger 2 length • 𝑑3: finger 3 length • 𝑏: body length • 𝑙: hind limb length Four Angle Parameters: • 𝛾1, 𝛾2, 𝛾3, 𝛾4
  26. 26. Optimization Iterative optimization of flapping parameters and kinematic parameters Iteratively optimize  flapping angle trajectory  spindle trajectory  link lengths and offset angles  hind limb trajectory Optimization Criteria: • Minimize Euclidean distance between selected makers on the biological bat and the corresponding locations on the robot bat.
  27. 27. Preliminary Results • Solid diamonds are biological bat data. • Open circles are for the optimized robot bat design. Future Work: • Use wing shape instead of marker locations • Use Principal Components • Parametric, periodic shape descriptors • Incorporate into robot design • Incorporate into trajectory planning
  28. 28. Concluding Remarks  Kinematic design – passive and active joints  Nonlinear dynamics – many unactuated degrees of freedom  Actuated degrees of freedom define a manifold on which the system dynamics evolve  Design actuator inputs to define this manifold, which shapes the system dynamics  Design a nonlinear control to track the desired unactuated dynamics  Preliminary results on 7 dof simulation  Simultaneous kinematics/trajectory design
  29. 29. Undergraduate Researchers Graduate Researchers Principal InvestigatorsPostdoc

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